Universal Enveloping Algebras

 

This is the main chapter of the essay, in which we introduce universal enveloping algebras. After the definition and showing some properties of enveloping algebras, we will have a section on the Poincarre-Birkhoff-Witt Theorem and prove it in full generality. This basically says that every (not necessarily finite dimensional) Lie algebra $\EuFrak{g}$ has a unique (up to isomorphism) enveloping algebra $\mathcal{U}(\EuFrak{g})$. This will be done by associating to each Lie algebra $\EuFrak{g}$ an associative algebra (with 1), which is generated as ``freely'' as possible by $\EuFrak{g}$ subject to the commuting relations in $\EuFrak{g}$. We will also show that there is always an embedding $\EuFrak{g}\rightarrow\mathcal{U}(\EuFrak{g})$ into the associative k-algebra $\mathcal{U}(\EuFrak{g})$, where k is an arbitrary field of characteristic zero.